We need to calculate a derivative of an implicit function. To do that, we use the chain rule in Derivative Rules on both sides of the equation and when we encounter y, we derive it, i.e. multiply by y’. This is how it goes:

y=\sin (3x+4y)

y'=\cos (3x+4y)\cdot (3+4y')

Now, we want to isolate y’. Therefore, we open brackets and move expressions with y’ to one side:

y'=3\cos(3x+4y)+4y'\cos(3x+4y)

y'-4y'\cos(3x+4y)=3\cos(3x+4y)

y'(1-4\cos(3x+4y))=3\cos(3x+4y)

y'=\frac{3\cos(3x+4y)}{1-4\cos(3x+4y)}

Have a question? Found a mistake? – Write a comment below! Was it helpful? You can buy me a cup of coffee here, which will make me very happy and will help me upload more solutions!